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RADC-TR-72-119 TECHNICAL REPORT APRtL 1972

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EFFECTS OF TURBULFNCE INSTABILITIES ON LASER PROPAGATION

RCA LABORATORIES

SPONSORED BY ADVANCED RESEARCH PROJECTS AGENCY

ARPA ORDER NO. 1279

Approved for public release; distribution u. Umited.

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The views and conclusions contain«! in this document are those of the authors and should not be interpreted as necessarily representing the official policies, eithir expressed or implied, of the Advanced Research Ptojtcts Agency or the U.S. Government,

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RCA Laboratories Princeton, New Jersey

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N/A i*. REPORT TITLE

EFFECTS OF TURBULENCE INSTABILITIES ON LASER PROPAGATION

{«■ DESCRIPTIVE NOTES C7yp«o/iwportan^Me(u«<r* *«••>!

Quarterly Progress Report No. 3 9 December 1971 - 8 March 1972, I «■ AUTHONltl (Pint namm, «»91 InlMaf, laat SSpJ

David A. de Wolf

6. REPORT

April 1972 7a. TOTAL NO. OP PACES

39 76. NO. OP REPS

11 •a. CONTRACT OR «RANT NO.,

F30602-71-C-0356 6. PROJECT NO.

ARPA 1279 c- Program Code 1E20

PRRL-72-CR-12

NUMBER«)

I d.

*b. OTHER REPORT NO(S| (Any otfiar niimbar« tfiatmay ba äääTänäd Mia nporl)

RADC TR-72-119 !

110. DISTRIBUTION STATEMENT

,Monitored by: Lt. Darryl P. Greenwood (315) 330-3443; RADC (OCSE) GAFB, NY 13440

II. SPONSORINO MILITARY ACTIVITY

Advanced. Research. Project Agency 1400 Wilson Blvd. ' Arlington, VA 22209

VCT

Several statistics of the focal spot of a focussed laser beam in turbulent air are computed. The average area is recomputed for horizontal propagation,iand corrected graphs are presented. Comparison to the half-power: ariea is made, and it is concluded that the irradiance-weighted radius variance is a slightly better criterion, although the differences are small. The calculations are extended to slant paths. The power spectrum of the atfea scintillation is shown to be1 proportional to that of angle of arrival of a ray. ! It is computed for frozen flow and for random flow w,itfy zero mean velocity. Both spectra are initially flat, then decay as f"2/3. A composite formula is suggested fo'r the general case.

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Wave propagation Turbulent atmosphere, air Laser beam, focussed Optics, atmospheric Irradiance fluctuations

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EFFECTS OF TURBULENCE INSTABILITIES ON LASER PROPAGATION

DAVID A. de WOLF

CONTRACTOR: RCA LABORATORIES CONTRACT NUMBER: F 30602-71C0356 EFFECTIVE DATE OF CONTRACT: 9 JUNE 1971 CONTRACT EXPIRATION DATE: 8 JUNE 1972 AMOUNT OF CONTRACT: $49,990.00 PROGRAM CODE NUMBER: 1E20

INVESTIGATOR: DR. DAVID A. de WOLF PHONE: 609 452-2700 EXT. 3023

PROJECT ENGINEER: LT. DARRYL P. GREENWOOD PHONE: 315 330-3443

Approved for public release; distribution unlimited.

This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by Lt. Darryl P. Greenwood. RADC (OCSE), GAFB, N.Y. 13440 under contract F30602-71-C-0356.

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FOREWORD

This Quarterly report was prepared by RCA Laboratories, Princeton, New Jersey under Contract No. F30602-71-C-0356. It describes work performed from 9 December 1971 to 8 March 1972 In the Communications Research Laboratory, Dr. K. H. Powers, Director. The principal investigator and project engineer is Dr. D. A. de Wolf.

The report was submitted by the author on 7 April 1972. Submission of this report does not constitute Air Force approval of the report's findings or conclusions, it is submitted only for the exchange and stimulation of ideas.

The Air Force Program Monitor is Lt. Darryl P. Greenwood.

PUBLICATION REVIEW

This technical report has been reviewed and is approved.

!ADC Project Engineer \»

11

ABSTRACT

Several statistics of the focal spot of a focussed laser beam in turbulent air are computed. The average area is recomputed for horizontal propagation, and corrected graphs are presented. Comparison to the half-power area is made, and it is concluded that the irradiance-weighted radius variance is a slightly better criterion, although the differences are small. The calculations are extended to slant paths. The power spectrum of the area scintillation is shown to be proportional to that of angle of arrival of a ray. It is computed for frozen flow and for random flow with zero mean velocity. Both spectra are initially flat, then decay as f"2/3, A composite formula is suggested for the general case.

iii

SUMMARY

Atmospheric turbulence causes the focal spot of a focussed laser beam to scintillate, i.e., the irradiance distribution in the focal plane varies irreg- ularly. In the first quarter of the contract period (9 June to 8 September 1971), a simple calculation of the variance of an irradiance-weigh ted radius of the focal spot was presented. It yielded an average focal-spot area equal to^he sum of the diffraction-limited area and an area proportional to LJKm Cn (L - range, icm - microscale wavenumber, Cn

2 - refractive-index structure

constant). There exist arguments whether such a variance represents an effective area

as well as the half-power area in the focal plane. Moreover, some numerical errors crept into the analysis and the graphs. Hence, the entire matter is presented again in Section I with errors rectified, and with a considerable discussion of the "differing" definitions of effective focal-spot area. We conclude that the differences are unimportant, and that the irradiance-weighted

variance is more proper. In Sections II and III, the analysis is extended to slant paths, ground-to-

air, and air-to-ground. The theory summarized by Wvngaard, et al.[5] is uti- lized for incorporating the height dependence of Cn2(z) into the theory. The effects are much less than in ground-level horizontal propagation, and weakest for the ground-to-air case. Graphs are presented that allow one to convert the horizontal-propagation parameters into slant-path equivalents, when altitude of

laser or receiver is known. In Section IV, the power spectrum of focal-spot radius scintillation is

computed. It is proportional to the power spectrum of angular fluctuations of a ray. We compute the latter for two cases: (i) Taylor's hypothesis, i.e., frozen flow with a steady wind velocity, and (ii) random velocity distribution with zero mean. An approximate formula is constructed for the composite of these two cases: it yields a spectrum that is a function of a square velocity v2 ■ UT2 + 16AU2/3. Here, UT is the mean-velocity component across the beam, and AU^ is the velocity variance. The spectrum is flat when u < v/L0 (u is the angular frequency variable of the spectrum, L0 is the macroscale). It decays as u"2/3 for v/L0 < u < Kmv, and cuts off quite sharply when w exceeds umv.

iv

TABLE OF CONTENTS

Section Pa8e

SUMMARY iv

I. HORIZONTAL PROPAGATION PATH 1

II. SLANT-PATH PROPAGATION: GROUND-TO-AIR 7

III. AIR-TO-GROUND PROPAGATION 12

IV. SCINTILLATION RATES: ANGULAR-FLUCTUATION POWER SPECTRUM 14

A. Introduction 1* B. Frozen Flow (Taylor's Hypothesis), AU2 - 0 16 C. Random Velocity (No Mean), Uj; - 0 18 D. Summarized Results and Approximated Power Spectrum for General

Case 19

APPENDICES

I. THE VARIANCE-WIDTH THEOREM FOR FOCAL-SPOT AREA 24

II. THE RANDOM-VELOCITY HYPOTHESIS 27

REFERENCES 29

DD FORM 1473

I , ?IW-,/-..

LIST OF ILLUSTRATIONS '

i i

Figure Page

1. The free-soace focal-spot radius r^, and Its average broadening rLB ^ ^'L ^ " 'Lo^LB ' a8 function8 of range L for

10-14 <cn2 m m"2/3 <10-12 ,. 4

2. Maximal aperture radius roc required to obtain a minimal average focal-spot area as a function of range L for 10-14 <cn2 in m-2/3 < io-12 at A - 0.6 ym. . ., . . .; . . . . . . 5

3. Maximal aperture radius roc required to obtain a minimal average focal-spot area as a function of range L for 10"14 < cn

2 In i m-2/3 <10-12 at X - 10.6 ym 6

4. Cn(z) In cm-1/3 vs altitude z; the "Hufnagel curve". The Wyngaard, et al. predictions [Reference 5] are included .!...., 8

i 5. Modification factors for rLo and roc (from Figures 1 to 3)

for ground-to-air propagation for laser transmitters at altitude Z ....... 10

6. Modification factors for r^ and r (from Figures 1 to 3) for air-to-ground propagation for laser transmitters at altitude Z 13

|

i

i i

i i

vl

i

I

I. HORIZONTAL PROPAGATION PATH I , . '

! ; i

In the First Quartetly Report[1], a formula was derived for the effective focal-spot radius rL of a focussed laser beam propagating through turbulent air characterized by strength Cn

2 and scales Ä0 « 5.92/icm and L0. The definition of rL is

2 rL 'jd\ p2 ,<i(p,L)>/yd2p <I(P,L)> a)

hence it is an average-irradiance weighted variance of the distance in the focal plane from thp central axis. The result we obtained is

1

1 . '

' r 2 - (L/kr )2 + 1.3 L3K 1/3C 2 (2) L o m n . v '

for a Gaussian beam. The general result is

2 = . 2^ , Q T3.c 1/3C 2 i 'm n

rL ^Lo +1-3LKm C (3)

where rLo is defined by (1) if < I(p,I1)> is replaced by the free-space Io(p,L). The result (2) is essentially the same as that of Varvatsis and Sancer[2].

i '

Lutomirski and Yura[3] define another effective focal-spot radius rL* (z8 in their notation) which is defined as a half-power radius such that <JI(?,L)> =■ 1/2 <1(0,L)> for p = n/. They obtain

r' « 0.9 L3/2 Km1/6C for 0.4k2LÄ

5/3C 2 » 1 ^ m n i i o n

1 , i (A) ^ 0.7L8/5k1/5Cn

6/5 for 0.4k2LÄ 5/3C 2 « 1

i n on

There is also a lower bound on L for the second criterion in (4) but it is of the order of one or several meters for systems with apertures of at least 10 cm diameter, and hence not of much practical importance.

i ' ''

i The results (3) and (4) appear different for L less than (O^k2^5/^2)"1, a distance of roughly 7 km at 0.6 ym and more than 2000 km at 10.6 ym. This difference thus appears to manifest itself in practical situations, particularly in the infrared regime. At the tjlme of writing - early 1972 - there appears to be iiis^greement among workers in the field about relevance and applicability of results (3) and (4). The purpose of this section is to clarify the matter.

The peculiarity of result (3) is the following: for truncated beams - particularly for the so-called Airy-disk case - one finds r. - «, hence rT - »

Lo L , i i

I , i I

1

' vHSvMlwBMHraHB

2 2 and the result appears meaningless. Moreover, r -r depends on the micro- scale Ä0 (or K -1). L Lo

m

The peculiarity of result (4) lies in the somewhat arbitrary definition of rL* as a half-power radius,* and the dependence of ri/ upon the macroscale L0.

We wish to make a case for the statements that result (3) is more funda- mental than result (2) - also in the case that rLo -*■ " - and that the differ- ence between (nZ-rLoO1'2 and 'L* is rather small for all practical cases.

In order to do so, it is first noted that the definitions of

(rL -rLo > ' - rLB» and 1'25 rL coincide when 0.4k2Uo ' Cn2 » 1- The

reason is that the irradiance profile at z » L has become Gaussian in which case the half-power radius differs from the irradiance-weighted variance by only a numerical factor ^1.25.

The next point is that (3) is a very general result, as noted on page 9 of Reference[l]. In Appendix I, we give a derivation well-known in radio astronomy[4] where an analogous problem exists in analyzing the power spectrum of backscattered radio pulses. The spectrum is a convolution of the spectra of the target scattering amplitude and the pulse envelope. Hence the variance width of the spectrum (which tells you something about the Doppler content of the astronomical object viewed) is the Pythagorean sum of the variance widths of the two component spectra in the convolution. The variance width of a square- pulse spectrum is infinite. Hence the analogy. The difficulty is circumvented by regarding the difference of the variances which can be computed in any prac- tical case by exploiting the finite bandwidth of the observed spectrum. In our case, that means that rLB2 " ^L^Lo2 can be computed in practice because rLo is not really infinite: it is limited by finite signal-to-noise ratios and if these are taken into account oonsistently for rL and rLo there is no real difficulty. I.e., one computes from the data,

pn

rL2 = / d2P P2 [<I(P.L)> -I0(P.L)| / I dZp <I(p,L)> (5)

where pn is a value of the radius beyond which the patterns vanish into noise. If the broadening due to turbulence occurs well within pn, i.e., if the result of the above calculation yields ^B « Pn» then it follows that the denominator of (5) is adequate since it is essentially equal to the total power emitted by the laser. It is quite true that the second p-moments of < I(p,L)> and I0(?,L) are proportional to pn, but their difference is not. The quantity ^B2 there- fore can be calculated and has meaning even when the variance width rL 2 0f the vacuum irradiance pattern is infinite. The irradiance pattern is thus broad- ened in turbulent air by convolution with the Fourier transform ^(q) - see Appendix I - of the mutual-coherence function of a spherical wave field in turbulent air divided by the equivalent vacuum field. The quantity ILB2 is simply the varianca width of this Fourier-transform pattern IOGJ), where <r-(k/L)ir. ^^

♦However, definition (II) of Ref. 1 yields essentially the same result (private communication, D. P. Greenwood).

2

I

In my opinion this settles the question of whether the resulting broadening should depend upon the macroscale L0 or upon the mlcroscale £0. The broadening Is clearly determined by the shape of IßCq)• The variance width rnj* of IßC?) Is finite and therefore clearly a valid measure of the width. It Is given by (A-10), hence It Is a function of the miavoaaale. It Is fairly easy to under- stand why the broadening depends more essentially upon the mlcroscale. The definition of IßG?) In (A-3) makes It perfectly clear that the radius parameter <t ■ (k/L)? of the Fourier-transform pattern In the focal, plane Is the Fourier- conjugate variable of the difference coordinate Ap In < B(1)B*(2)> Consequent- ly the behavior of Ijjfa) for large q Is determined by the small Ap-behavlor of < B(1)B*(2)> . These are both therefore determined by the miorosoale l0. On the other hand, the maovoaoale L0 determines the behavior of < B(1)B*(2)> for large Ap, hence of IB(<D for small q.

What we are saying is in effect that the half-power radius r^' is not really a good measure of the broadening (although we will show further on that it isn't bad either). Remember that the pattern <I(p,L)> in the focal plane is a convolution of I0(p,L) with IgCp.L) ■ IßC^l» When In is broad compared to I0, obviously the broadening is given by r^g^ ^ L^<n ''cn^ and the first criterion for r^' given in (4) must hold. When Ig is narrow compared to I0 the broadening is also given by rLg 'V» L Km^'^Cn^, but this quantity is then small compared to the width of the main lobe of I0, and therefore not interest- ing. The interesting case is when rLg is comparable to the main-lobe width in which case Lutomirski and Yura[3] utilize ri,' ^ L8/^1/5^6'5 and the second criterion in (4). At Infrared frequencies, most propagation paths of Interest fall into this case when Cn2 ^ 10~^m

-2' . What happens in this case is that the nulls between main lobe and side lobes of IQ are rapidly filled. The total patterns < I> given in [3] indicate this at prths L larger than several hundred meters (L = 100 zc). The half-power radius then becomes rather arbitrary.

There is another way to examine this difference between rj/ and rjjj. It is simply to compare them! In fact, let us compare r^g and 1.25 r^' to each other, since these coincide for a Gaussian beam.

1.25 rL' W'V! . / -5/3A c 2VA0 (6)

m n ^•^äTTTütr-w tLC^

For fixed frequency, this ratio varies with (LCn ) , i.e., very weakly. Clearly, the definitions are not very different. More serious, however, is the restriction of validity of the numerator of (6) to the condition (p. 1657 of[3]),

10 < 0.4k2L L 5/3 C 2 < 104 (7) o n

for errors of less than 25%. By taking the 10-th root of (7), we obtain

mmmmmmMmmmmmmmmmmmmmv''' ■'mKmKmtm''

1.4(K L )'1/6 < 1.25T' /r^ < 2.80c L )"1/6

in o L LB m o (8)

A typical large surface value for i^L,, is 10 , hence we note th..;- in the region

'i' of validity of the expression for rT', we have

0.4 < i'25^ /rLB

< 0.9 (9)

Relationships (6) and (9) indicate clearly that there is not much numerical difference between rj/ and rLB in the region of validity of the cited expression for ri/. On the other hand, the expression for rjLB is valid over a wider range. Hence it is concluded that (3) is to be preferred to (4), although careful use of (4) does not lead to appreciable differences.

2 Horizontal propagation is characterized by constant Cn along the propaga-

tion path. It is therefore fairly simple to plot (3) for this case. Figures 1 and 2 of Reference[1] contain some inaccuracies, hence new graphs are sup- plied in this report.

100

10

E u

0.1

rL

Cn2 in nr2/3«IÖ

Figure 1.

The free-space focal-spot radius rio» and its average bro iening rLB «rL2> = W+fLB2) as fac- tions of range L for 10"1* < Cn

z

in m-2/3 < io-12.

In Figure 1, we plot rj, vs L according to (3). rLB> t^e second term In the right-hand side of (3). r^Q for two different frequencies.

The solid lines represent The dashed lines represent

In Figures 2 and 3, we plot roc vs L. The critical laser aperture roc Is defined to be such that rLo ■ 1.8 • If roc is Increased, then rx.0 Is decreased and rt ^ rLB. That Indicates that larger apertures than those with radius roc will not yield smaller average focal areas: a lower limit irruj2 masks the diffraction area. By setting rL0 = L/kroc equal to ri^ we obtain

oc 1/1.4 kL1/2 K 1/6 C m n

L/kr. LB (10)

E u

u o

1 1 r-

^s^s>^ X»0.6/i.m

10

^\^ Cn2 in m^^-IO-14

1

1 1 1

10

Figure 2. Maximal aperture radius roc required to obtain a minimal average focal-spot area as a function of range L for 10-1* < Cn2 In m-2/3 < io-12 at X = 0.6 um.

"

100 -

E o

u o

Figure i

3. Maximal aperture radius roc required to obtain a minimal average focal-spot apea as a function of rarige L for 10-14 < c„2 m m-2/3 < io-12 at A .\0#6 ym>

6

i i

i , .i

i II. SLANT-PATH PROPAGATION: GROUND-TO-AIR

i ' ■' . ' !

In this section, we will modify (3) for the case of slant-path propaga- tion in which case Cn

2 is a function of altitude. To do so, we note that the main difference with horizontal propagation is tha,t instead of obtaining in (A-10) a factor

1 ■;■,' I I

' a': i

, Le2 J dKK3*(K) i i 0 ! ^ i ' ! , ^

i ■ : ,

from the limit p ->■ 0 in (A-9), we should be obtaining

lim p->0

*-*J dse (s) / dKK*(K)[l - Jo(Kp8/L)];

JU

f dee2(a) (s/L)2 f dKK *(K)

(11)

■ I

for a slant path characterized by a path parameter s. Equation1 (11) inserted ,into (A-10) yields

L: i

rL2=3,9L2K

1/3 f ds(s/L)2 C2(z) m / i nv / (12)

In adapting (12) to a situation where the laser is located at an altitude z0 close to the ground and pointed upwards at an elevation <anglä of 6 with the horizon towards an object at distance L, it is noted that path coordinate, s is related to altitude z through s = (z-zo)/sin9.

2 The behavior of Cn (z) with altitude is fairly complicated - particqlarly

at altitudes above 1 km. The Hufnagel cu^ve. Figure 4, indicates the diffi- culties: occurrence of random disturbed layers and dependence of lower- altitude Cn2 upon time of'day. Wyngaard, et all. have surveyed recent develop- ments in altitude dependence of Cn

2[5]. Their results yield.

Cn <Z)'" Cn ,(Zo)(T0/T)2(zo/z)

2/3 (l+z/£8)T2/3

x exp[-2(z-zo)/h] , (13)

Wi t : n

Ill IIIIIIIIIIIMWIH IWIIJJWow——w

SUNNY DAY

(HEIGHT)'

DISTURBED LAYERS

CLEAR NIGHT

(HEIGHT) •2/3

ü Im 10m 100m ikm 10km IGQkm

ALTITUDE ABOVE LOCAL GROUND

-1/3 Figure 4. Cn(z) In cm vs altitude z; the "Hufnagel curve".

The Wyngaard, et al. predictions [Reference 5] are included.

where T is the temperature at altitude z (T0 at z0), h is the scale-height of pressure at sea level (h « 8.3 km), and Ä8 is a length parameter related to the Monin-Obukhov length Lm by Äs - -LMO/7. This model predicts (for 2z « h).

-4/3

C 2 - z-2/3 n

sunny day - midday (l 1.5 m)

dawn, dusk (i -*■<*), s

and Figure 4 bears the prediction out quite well. The older Fried model, cited by Titterton[6], and Sutton[7] will therefore not be followed. Note that Equation (12) is a function only of the receiver altitude Z - LsinG, and that it can be rewritten as

rLB(Z)-rLB(0)-F(Z>

F(Z) -| f dz exp(-2z/h)(z/Z)2 (14)

x (1+Z/Zä)~2/3

[1+(Z+Z )/£ ]"2/3 o s

Note that the temperature dependence on altitude has been ignored. It is variable, but weak compared to the other s-dependent parameters in the F(Z) factor of (14). Certain approximations simplify the calculation of (14) :

(i) sunny day - midday situation: We may replace the last two fac- tors in (14) by (zoils)2/3 (z)"4/3 provided Z » z0,ls- For Z « z0 we obtain F(Z) - 1. The critical altitude Zc for which Zc ■» ZQ, Äg is of the order of 1 m or less when z0 « 1 m, and L > 1 km. Hence, slant paths are handled easily for most non-zero angles 6 by means of this simplification.

(ii) dam, dusk situation: Because Äs -*■ » in this case, we replace the last two factors in (14) by (z/z0)~

2/3 under the proviso that Z » z0, which is always true for any airborne target under consideration.

The remainder of this section is devoted to straightforward numerical expression of (14). It is easily seen from (10) and (14) that

roc(Z)/roc(0) = rLB(0)/rLB(Z) = [F(Z)]"1/2- (15)

Hence, it suffices to plot (15) as a function of Z in order to infer the slant- path quantities from those at zero altitude for an equivalent path of length L.

(i) sunny day - midday: With the previously discussed simplifica- tions, it follows that

F(Z) * 1.8 [(z A )1/2/Z]4/3 for Z « h/2 o s

(16)

-x- 0.85 h5/V * )2/3/Z3 for Z » h/2 , o s

the intermediate regime is governed by the incomplete gamma function from which (16) is derived.

(ii) dawn, dusk: Likewise, we obtain

f

F(Z) ^ 1.3(z /Z)2/3

.O.ThH^/Z3 o

for Z « h/2

for Z » h/2. (17)

In our view, the Z » h/2 forms in (16) and (17) should not be regarded very seriously because they depend upon validity of the model (13) above z = h/2 'V' 4 km, and that is certainly above the altitudes where this model is valid.

In Figure 5, we plot the Z « h/2 forms of [F(Z)]"1/2 vs Z, using Ä-o « ZQ « 1 m. The plots are substantially the same as those in the Second Quarterly Report (9 September - 8 December 1971, sequel to Reference[1]). However, it is simpler to plot the ratios as functions of altitude Z, and to modify the horizontal-path results of Figures 1 to 3 by these altitude factors when the image at the end of path L is at altitude Z.

10 100

roc<Z)/'bc<0)''LB(0,/rLB(z) —

Figure 5. Modification factors for rL0 and roc (from Figures 1 to 3) for ground-to-air propagation for laser transmitters at altitude Z.

10

Clearly, the effect of turbulence is much less pronounced on slant paths. It should be stressed that the curves In Figure 5 depend on z0l'3 and on Ä8^'3t and that we have chosen z0 « 1 m, ls « 1 m. The slant-path results of Figure 5 should be scaled accordingly, if different values for z0 (laser height) and £8(- -IMO/7) are utilized. Note that Z - L sin 6; these curves are functions only of the product L sin 6. Each point describes a variety of path- lengths L and elevation angles 6 subject to L sin 0 » Z.

li

III. AIR-TO-GROUND PROPAGATION

If it Is the transmitter that is at an altitude z - Z, and the receiver at z - z0 such that the propagation path has a length L at an inclination of -9 with the horizon (ignoring curvature effects), then a slightly different calculation ensues. Instead of (12), we obtain

r* = 3.9 L2K1/3 f ds(l - s/L)2 C2(z) (18)

0

Because the dominant part üf the integral is in the first 150 m or so above the ground, where (1-s/L)2 M 1 for altitudes Z of 500 m or more, and because the model for Cn

2(z) is not trustworthy above several hundred meters, we may just as well ignore the factor (1-s/L)2 in (16) and adapt (14) to the present case by writing

rLB(Z> -rLB(0) F,(Z)

Z (19> F,(Z) = | / dz exP(-2z/h)(1+z/z0)~

2/3[l+(z+z0>/Äg]"2/3

0

We have utilizod LsinS = Z in this formula, as in the previous section. The numerical work is a little more complex than in Section II.

(i) Daum, dusk: In this case (17) simplifies to

3z Z/Zo

F.(z) = -^- J dx (14x)"2/3 exp(-2xZ/h),

1

and when Z « h/2 (we need not consider the other case because the model is then too uncertain) it follows that, to good approximation,

F'U) % 9(z /Z)2/3 for Z « h/2. (21)

(ii) Sunny day-midday: To sufficiently accurate approximation, (19) can be simplified to

(20)

1/3« 2/3 Z!Zo F,(z) m 32° - -S J dx x-2/3(l-fx)'2/3exp(-2xzo/h),

1

and a further approximation is to replace Z/z0 in the upper bound of integration bv ». Replacing x by y-1/2, we obtain an integrand (y2-l)"2^, in which we may ignore the -1 term; i.e.,

(22)

12

F'CZ) ^

3zl/3£2/3 5! O 8 /dy y-4/3 ^ 9 « 1/3Ä 2/3/Z. ' OS

(23)

3/2

The resulting expressions (21) and (23) are Inserted Into (15) which Is plotted li Figure 6 for air-to-ground propagation. The effects of turbulence are somewhat larger In this case than In the equivalent opposite case: ground- to-air. The reason Is that the filter factor of (12) eliminates contributions from low-altitude Cn

2(z), whereas that of (16) removes hlgh-altltude (much lower) Cn

2(z). This may appear surprising, but we must remember that the focussed bundle consists of spherical wavelets leaving the aperture at z = 0 and coming together close to z - L, each at Its own angle. Turbulence affects the direction of a spherical wave most when the distance from the point of transmission Is large. The effect Is analogous to the "dirty windshield" effect: a driver with a dirty windshield can be blinded by the headlight of an oncoming car, but the oncomer cannot see that the windshield of the other car Is dirty.

10

I -

N

i 1

SLANT PATH J DOWN

/ / / / / / / / / / 1

1 /

/

- i . / -

*l Oil

7 7 1

^ 10 100

roc(Z)/roc(0)-rLB(0)/rLB(Z)—*

Figure 6. Modification factors for rL0 and roc (from Figures 1 to 3) for air-to-ground propagation for laser transmitters at altitude Z.

13

wmimmmmmmBitmmmnam

IV. SCINTILLATION RATES: ANGULAP.-FLUCTUATION POWER SPECTRUM ,

A. INTRODUCTION !

l

So far, we have discussed only the average focal-spot broadening due to atmospheric turbulence. What really happens is that the focal spot scintillates around the average. Consequently the next questions deal with higher-order statistics such as the variance of the focal-spot area around the mean, and with the time scales on which this phenomenon occurs.

i

We will occupy ourselves in this section with the time scale of Scintilla- tion; i.e., with scintillation rates. There are many ways to define a signifi- cant measure of scintillation rates, but the one most commonly used is the pcwer spectrum of a scintillating quantity. The power spectrum is the Fourier transform (with respect to separation time T) of the autocovariance of the scintillating quantity with itself at a time T later. It is a function of the conjugate variable u, an angular frequency. ' ! i

Let us try to define an appropriate autocovariancfe function RTR(T) that- reduces to rLB^ when x -► 0. To do so, we note that the average of Equation i (A-l) is a special case of ,

<E(r;t)E (r;t+T)> i f O / \

" /d2pl/*2p2 <B(?'?rt)B*^'?2;t+T)>Uo(?l)üo(?2)exP[ikP-(PrP2)/L]-

When T -»■ 0, we note that the above formula reduces to thai for < I(r); > . So. the new definition that we Introduce for RLB(T) is obtained from (5; by re- placing <I (?r,L) > in the numerator by (24), viz..

00 ^

^(T) - J d2pp2[<E(?;t)E*(?;t+T)>-Io(r)]/ fd2p <I(r)> , >

i i

(25)

where r « (p,L). In fact, the numerator of this definition can be obtained by multiplying p [E(r;t)-E0(?;t) ] by the complex conjugate of p [E(?;t+T)-Ena;t+i)]. averaging this and then integrating over the focal plane (the cross terms are zero on the average). It is therefore plausible that' R(T) is physically, analogous to the autocovariance of a focal-spot radius at time separation x. Let us therefore adopt (25) as the desired autocovariance. i

.o/x V thetderlvation in Appendix I is followed with this one difference: i

:,^*s0 USed in8tead of (A-2). it is readily seen that we obtain, instead or (A-10), i ■

I

2 00

h*™-!^ /dKK3*4i(K;T). (26)

1 2 ' 1 ' ' : ->■ -► where e ^(K;!) is the Fourier transform of < 6e(r1;t)6e(r2;t+T)> averaged also over t, with respect to thp difference coordinate ^i-^ ■ Ar. It will be noted by comparing (26) with Appendix II of Reference [1] that

i 2 '

^(T)-^- < 6^(t)«6?(t+T)> , (27) i

i

I !■ i.e., that the autocovariance of a weighted focal-spot radius is intimately associated with the autocorrelation of the angular deviation 6t of a ray over a path of length L. For those who feel uncomfortable with the definition given for RLB(,T)» we can therefore state that another physical interpretation of this quantity is that it is proportional to the autocovariance of the angular deviation of a ray. Let us therefore define

i Re(T) = < 6t(t)»6t(t+T)> i (28)

: 'it A2KK2VK;x)., , i i

The qonnection between Rg and RLB follows from (27). We shall now restrict ourselves to, (28), and to horizontal propagation with uniform Cn

2 along the path for simplicity. Extensions to slant paths will be pointed out where germane. . ! •

The major point in further developing (28) is to make a good hypothesis for ^(K;!). By definitjlon ,

*4(K;T) = fd3Ar <6e(r1;t) 6e*(r2;t+T)> e1K,Ar, (29)

where averaging oyer time is implied. In Appendix II, we utilize Favre's hypothesis[8] to find (A-17) as the major step in ensuihg development. In- serting (A-17) into (28), one obtains,'

i 00

Re(T) '%-■ J dKK3 *(K) J0(KU

T T) exP<- jK2AU2T2), (30)

I , ! ■ 0

where Ux is the component of the mean velocity normal to the propagation direction, and AU? is the variance of the wind velocity In Isotropie turbulence. When &U2 - 0, Equation (28) describes "frozen flow" or "Taylor's hypothesis": the motion is a steady convection 'of a random1 spatial structure of eddies (that are immobile with respect to each other) across the beam with velocity

15

component Uj. By using Equation 1-3 of Reference [1], and transforming K to a new variable x ■ K2L02, we rewrite (30) as

CD

VT> - ^ * £ / d-<1+*>'11/6 Vv ^ 0 0 (31)

x exp[-x(Atü T /4 + 1/K L )], mo

where u^ - UT/LQ and Au2 - 16AU /3Lo are the frequencies obtained by dividing the previously Introduced velocities by the macroscale L0. Finally, the power spectrum of angular fluctuations, WQCU), is the Fourier transform of (31):

00

W (u) = 2 J dx Re(T)cos ü)T (32)

0

A simplification can be made by noting that x(14x) = (1+x) -(1+x) It allows one to rewrite (31) and (32) as

W («) = i|^ . iJi [We(u),5/6) -We(u),ll/6)], (33) o

with the following definition of W (w,q):

00 oo

we(u,q)' f ^ f dx a+«)'q J0(V ^

x exp[-x(Au2T2/A + 1/K2 L2)] COS UT, (34)

Thus, we attempt to compute (34) for q - 5/6 and for q «= 11/6, and Insert the difference into (33) to obtain the power spectrum. Unfortunately, the double Integral In (34) does not appear integrable into a tabulated function in general. However, we can do two special cases, namely Au B 0 and CUT = 0. These two cases are the subject of subsections B and C.

B. FROZEN FLOW (TAYLOR'S HYPOTHESIS), AU2 - 0

In this case, it is easiest to carry out the Fourier transformation in (34) first, inverting the order of integration, to obtain

00

We(W,q)-y* dx (14«)"q(x(1»2-üi2)"1/2 exp(-x/K2 L2). (35)

2/ 2 w /uT

16

imwmmmm

Let us also Introduce the frequency OT " ^m^T» lt corresponds to a velocity divided by a microscale. Equation (35) is easily transformed into,

W-,, - ^ e-»2/^ / ^1/2<^2 + 4'-q ' ■"*

2,.2 / 2, 2 <3'a)

The second form was obtained by utilizing Equation (14.4.12) of Reference [9], and converting the Whittaker function in the result to a Kummer function U(a,b,z) as in Equation (13.1.33) of Reference [10]. Let us see what W^u.q) looks like for different parameter regimes:

(i) u « IL,:

For the case that u is much less than the microscale-defined frequency ftp '^ UT/£0, we can utilize Equations (13.5.10) and (13.5.12) of Reference [10] to obtain

„ , , r(q-l/2) /H /, . ü)2\

1/2"q ,,,., VU'^ "TOO ^ ^1 + 7J (36b)

(ii) ü) » nT:

In this case, we utilize the asymptotic form of Rummer's func- tion U(a,b,z) ->■ z-a to find.

V"'-» «A ^ f1+S) "q e ""^ • (36c)

This formula is not very interesting since u » fi-p implies a strong Gaussian decay factor making (36c) quite negligible compared to (36b) .

Note that the power spectrum is flat below u« UT/L0, and then decays as w"2/3 for Ux/L0 < w « 11^/^. As u - approaches U-J/ÄQ , the decay sharpens -into a Gaussian cut-off, but this has as much meaning as the Gaussian cut-off in *(K).

17

fp»ÄÄ<wwii^4i(<w.^^>tiiP',i-"'»w|,wt!|MA^öR*l^^s'''C,!;'WWBw:*' 0'W<W"~ '•""'' JWUrÄW""" ''*" *"*

C. RANDOM VELOCITY (NO MEAN), U - 0

Returning to (34), we again carry out the Fourier transformation first to obtain,

V'-O " i / d- 1/2(l*0-q exp -U^ - -fa] . (37) rt VxAu) K L / 0 ^ m o/

This case Is slightly harder tc handle than the previous one. We can drop the x/<m2L0

2 term in the exponential because the first term in it will take care of convergence. With thi simplification, we set x - 1/y in (37) to obtain

• ■'s

MM

W9(ü,'q)* 55" / dyyq~3/2 0-*y)"q exp(-yu)2/Aü)2)

0 (38a) 2\

The development of the second form is the same as that in (36a). It can be seen that the general shape is similar to (36b) by using the asymptotic forms of the Kummer function in (38a):

Wö(ü),q) ^ r^"l/2^ —■ for u « At».

6 r(q) Au

■v r(«l-l/2) £ ^ q for a, »co. (38b)

We can also develop another form from (37) for u » Au by retaining the exp (-x/icm

2L02) but setting (l+x)-? * x"«!. Equation (37) is then reduced by

utilizing the Laplace transform (4.5.29) of Reference [9], and by Introducing the microscale-governed frequency AfJ2 ■ l6Km2AU2/3. We obtain (for q > 1/2),

1/2-q v_ { m o j

\ Au /

* T«-l'V § fa) q for « « Afi (38c)

* (,cmLo) fc (In) «P(-2"/A«) for « » ^.

18

Here, Kq.j^CZw/AQ) Is a modified Bessel function In the notation of Reference [10], and the two asymptotic forms are easily obtained. We note that the second form In (38c) agrees with the second form In (38b) thus confirming the validity of the approximations. It is also clear that the power spectrum cuts off abruptly as u> approaches and exceeds Aft ^ AU/H .

Just as in case B, we note that the power spectrum starts off flat below m * Aü/L0, and then it decays as üI-2/3 until the cut-off at u *« AU/£ .

D. SUMMARIZED RESULTS AND APPROXIMATED POWER SPECTRUM FOR GENERAL CASE

The results found in this section aaiount to the following. We assume a random velocity distribution with mean U0 and variance AU

2. The component of mean velocity across the beam is U?. We have defined a power spectrum of focal-spot radius fluctuations that is proportional to the power spectrum of angular fluctuations. The latter is named WQOü). It has to be a function of Uj and AU2. We have defined some frequencies by dividing these velocities by length scales:

w- - UT/L , Au2 - 16AU2/3L2

(39)

0^ - KjU.j, , Aft2 - 16K2AU2/3 .

Furthermore, we have found that WgOii) is the difference of two parts, see Equation (33). In Tables I and II we summarize the results for Wg(ü),q), bearing in mind that we subtract the q = 11/6 equations from the q = 5/6 ones to get the power spectrum.

It is Interesting to compare the random-flow with the frozen-flow results. The spectra are very similar. We would like to suggest an approximate formula for engineering purposes. To do so, we note from (31) two properties:

Re(0) - i / dWWe(a,) - ffjl ^ r(ll/6) (^L/'3 >

(40)

-[ 32Re(T)

3T2

T-0

I I A 2« / ^ 15.7 ezL r(7/6) , . .7/3 , 2 ^A 2. " W ^ We(a,) " l^T * L 2 ^mV (UJT +Au) >

« o

The second property expresses the variance width of the power spectrum. It is equal to the mean square width cause W^u) is symmetric around u = 0 so that we utilize cosine transforms in the preceding. The first property simply recalculates the angular-fluctuation variance and states that it is the zeroth moment of We((ü). The Interesting point is that (40) is a function of uj^Au2. We already know that the flat part of Wgto) (for u < ux, Au) does not contribute appreciably to Re(0) because it can be seen readily from (31) if we set 1+x - x

19

|:M 'HI

<

u > HJ O

H

I

«M

CM H

J 3

a" I

CM

CM H a

3 r

CM

CM H 3

CM

^ir

CM 'c^

CM

CM H 3 ,

CM

^3^

CM H c:

CM

■cr i CM

CM H 3

CM

^|3H

CM

I cr

cr w t-.

t 3

:M

a8" CM

cr CM I

3 3

O V

II

E-

cr (M

I

!>= 3| 3

^|3H

CM

a t-.

cr H

^3^

CM

er t

f-4

20

wmmmm

o

a i < IM

CM 3 <

3 CM

CM

il

^J

3 CM

CM

« I

I CM

3 <

CM

4:1 B'4

c: < 3

CM I

4>

^|§

1^

CM

3 3 I l<

^1 3 <

CM

l-i «N

I

3

< e

<?

a <

<

3 <

3

21

that the resulting power spectrum then has no flat part, and we have already seen that this Is a valid approximation in computing Re(0). Consequently, the flat part is even less Important in computing the second of (40). Since most of the area under WgCu) lies in the a)~2/3 portion (q - 5/6 yields an ü)~2/3 behavior that dominates over the q - 11/6, a)~8/3 part), it appears to us that this portion of the curve is governed by a parameter u)2/(urf2+Äu)2). Consequently, we suggest that the following formula is useful over the entire range of tai

2 We(a,) -i|jll3l. ^((0,5/6) - We(U,ll/6)],

o 2 v i/2^

exp[-(ü2/(^+AJ22)], (41a)

x ü[l/2, 3/2-q, (u2+wT2+Aa)2)/(n2+AQ2)] .

This formula clearly satisfies (40) because it reduces to (36a) if we set Au - 0, and (36a) yields (40) with Ao) ■ 0. It is obvious that by substituting ü)-];2+AUJ2 for ü)-J;2 in (36a) we satisfy (40) in general, and we obtain the above formula (41). When u2 « fJT +Af2

2, we can utilize the approximation

„ / x - 1^-1/2) / « | 1/2 /. . ü)2

W0(u,q)* ^,1 / -x- e^'^~ r(q) {2~2 r XT2

1/2-q

(41b)

Comparing (41b) to the results in Tables I and II, we note that we obtain the previous results to reasonable approximation. If we allow wf -► 0, we note that the (41b) approximation is in error for u << Au by a factor /n, and by a factor r (5/6) «1.1 for u » Au. This appears sufficient for engineering purposes. Furthermore, (41b) is in error when ux ■*• 0 for u Z An, but here the spectrum is very small and its precise shape is governed by the dissipation range of turbulence for which the precise shape is not crucial anyway. By choosing (41) as we have, we at least assure ourselves of the proper normaliza- tions (40) although the cut-off shape at u ^ Aft is not described properly. The transition regions are probably also imprecise.

Therefore, in conclusion it appears to us that the power spectrum of angular fluctuations, hence that of focal-spot scintillation, is adequately described by (41). That is to say.

22

mm

2 2 (i) the spectrum is flat for u < (ü)T +Aü) )

2 2 2 ~l/3 (ii) it falls off as [u /(u)T +Au) )]

for (a)T2+Au2)1/2 <"< (nT

2+An2)1/2. 2 2

(ill) it cuts off sharply beyond oi - (nT +AJi ) 2,1/2

To our knowledge, no calculations of We(u)) exist in the literature for this general case. However, Lawrence and Strohbehn[ll] cite untranslated work by A. S. Gurvich. M. A. Kallistratova, and N. S. Time (Izv. Vyssh. Uc.heb. Zaved. Radiofizika 11, pp. 1360-1370, 1968) for the Taylor's hypothesis case (Au - 0). Figure 10 of Reference [11] shows a plot of essentially a» We(u) vs. u which appears to be misinterpreted in the text of Reference [11]. All three portions (i) - (iii) of the behavior mentioned above are borne out by thJ8

graph which appears to us to correspond to u)x ^ 1 Hz (note ^^"T " 2lT l ^» and «T ^ 300 Hz (because the peak of the curve is at ic*T ^mL0fT * 50 Hz).

23

APPENDIX I

THE VARIANCE-WIDTH THEOREM FOR FOCAL-SPOT AREA

The purpose of this Appendix Is to derive (3) in a very general fashion and to stress the approximations in it. These do not appear to be understood as clearly as is desirable in context with the derivation in Reference[1].

Referring for notation to Reference[l], and specifically to Section III, we note that the irradiance in the focal plane (for a focussed laser beam) is,

lit) cc |yd2p1 BCr.r^ U^) exp(ik p-p^L) | 2 (A-l)

Here, ^(r^ is the transmittance amplitude, the so-called pupil function, of the aperture at z - 0, and coordinate rj = (PiO). Previously, UQ was chosen to be Gaussian. At present, it can be considered any generally accepted shape. The "sharp-beam" choice V0(ri) - 1 for pj^ s r0, and zero otherwise, yields the Airy disk in (A-l) when there are no atmospheric effects (B = 1).

The approximations in (A-l) are:

(i) radiation-zone, kL » 1 (ii) near field of the aperture, L « kr

(iii) sagittal, kro4 « L3 0

These are not very great restrictions at all for X - 10.6 ym and higher- frequency propagation. At 10.6 ym, one finds kr02 ^ 100 km, and (kr0

4)l73 ^ 30 m. Clearly 1 km < L < 10 km yields correction terms to (A-l) that are quite small.

To proceed further, we note that the expression inside modulus bars in (A-l) is a two-dimensional Fourier transform of a function of p1. The conjugate variable is kp/L - q, and consequently the irradiance is a function of L and of q. Therefore

Kq) - iy*d2p1 iKr,^) ü^) expdq-p^l2. (A-2)

When B(r,r1) = 1, we obtain I0(q), the free-space irradiance pattern in the focal plane z = L.

Only one more approximation will be made: it is assumed that every vector

r"rl for lr-rll ^ TQ i8 Perpendicular to the plane z = 0. This yields.

(iv) small-angle approximation, r << L

2A

MM

This approximation yields extremely umall errors and is easily satisfied in all

practical situations. It is needed only so that <B(r,ri)B*^?,?2)> can be re~ placed by the mutual-coherence function (mcf) with axis ? J_r]-r2.

Because < B(r,r1)B (r,^) > is a function of the difference coordinate Zp - rj-?2» * Fourier-transform pattern IgCl) can be defined as follows

<IB(q) =/d2Ap<B(r,r1) B*(r,?2)> exp(iq.Ap). (A-3)

The average of Equation (A-2) can be rewritten as a convolution integral of IB and I0 because of this property of the mcf of the normalized spherical- wave field B:

< V*» - 72 fd\ h^ V*-^ • (A-4)

This formula for the average irradiance is not new, but it is useful to stress its convolution property. It states that the free-space irradiance pattern I0(q) is broadened by the pattern IgCq) which depends only upon the mcf of normalized spherical-wave fields in turbulent air.

2 2 We now define variance widths for < I>, I0, and Ig, namely a , o0 , and

a„ respectively: a

2 _ a =

12 2 -*■ f 2 •*■ Jdq q" <I(q)> J d q q < I(q) >

/d2q <I(q)> \jd2q < I(q) > (A-5)

2 2 Likewise, definitions for a and Og can be written down, although omitted here. If (A-4) is inserted into numerators and denominators of (A-5), one finds the variance-width theorem after some rearrangement of terms:

2 2^2 ,A ,N o ■ a + o_ (A-6) O B

Note that the derivation depends only upon the validity of (A-2), which requires assumptions (i)-(ili) and the fact that the mcf < B(1)B*(2) > (in shorthand notation) is a function of ?i-?2' In applying (A-6) to effective irradiance-pattem widths, it is also noted that each a2 need be divided by (k/L)2 to yield actual widths. Thus, if we define r - (L/k)o, r - (L/k)o , and rL_ - (L/k)o_, we obtain

25

2 2^2 ,. _. rL ■ rLo + rLB ' (A-7)

For cyllndrlcally symmetric beams ql (q) is anti-symmetric with respect to the origin, hence the first moment of 1-13) is zero. The same holds for the first moment of I^ct). Hence rL2 is properly defined by (1), and rLo2, ry,

2 are defined likewise by replacing < I(p,L) > by I0(<J",L) and IB(p,L) - IB(q) respec- tively. Finally, it is noted that an expression for r 2 is readily obtained from the definition,

2 rLB

-yd2p p2 IB(q) /y*d2p IB(q), (A-8)

by utilizing Equation (10) of Reference[1] for the mcf. Unfortunately, part of the exponent in that equation is in error. The correct form for < B(1)B*(2) > with ? - ri-t2 is.

2 <B(1)B*(2)> - exp |- 3^ f dKK*(K)ri - fdz Jo(Kzp)j > (A-9)

When this is Inserted into the definition of Ig(q), which is then inserted in turn into (A-8), one obtains via a development analogous to that in Section IV of Reference[1],

2 L3 .2 /* 3 3 1/3 2 rLB - ^IäW <iKK3*(K) « 1.3L3Km C^. (A-10)

26

mm

APPENDIX II

THE RANDOM-VELOCITY HYPOTHESIS

The difficulty in (29) is that the autocovarlance < 6e (rj^t) 6e (r2;t + T)> depends on the time separation T. We will discuss a simple hypothesis that enables us to relate this autocovariance to the T - 0 case. Consider 6e(r;t) in turbulent air, it is proportional to the air-density fluctuation 6N(r,t). The continuity equation states that

dN/dt + NV«U - 0 (A-ll)

where N - N0 + 6N, and 5 is the gas velocity. Thus (A-ll) describes the time change of 6N because we^inay assume that dNg/dt - 0. In a hydrodynamical^df- scription, both 6N and ^ are functions of r and t, and dN/dt - 3N/ 3t + U'VN. The second term of (A-ll) is related to expansion or compression of the gas and the first term simply expresses conservation of density in a unit parcel of air as it flows from one location to another (perhaps changing its shape).

If we assume that "sources" of turbulence do not lie directly in or close to the propagation path, then we may assume Nv»U « 0 and consider the turbulence adequately described by dN/dt - 0, i.e., by flow governed by a wind-velocity vector Ü(r,t). It is probably not a very wrong assumption to thus ignore com- pression and expansion of the gas, and it simplifies matters because it implies that 6N(r,t) can be written in general as

t + T

6N(r,t) - 6N(r +/'dt, Ü(r,t '),t) (A-12)

t

and similarly for 6e «SN. It is easily seen that (A-12) implies dN/dt - 0 by allowing x -► 0 and keeping the first-order Taylor-series terms. These yield 36N/3t + U'^öN - 0 which is identical to dN/dt » 0 because No does not vary. We can apply (A-12) to (29) to obtain:

t + T

6e(r2; t + T) - 6e(r2 - f dt' Ü^t'); t) (A-13)

t

If this is inserted into (29) we obtain after a coordinate transformation: t + T

[iK-Ar» + Tdt' tl(r2,t')] (A-14)

♦4(^;T) -</d3Ar,6c(r1

,;t) St^it) e t >

27

where Ar' - tj^-tj1! r^' ■ r^, and r^' is the spatial coordinate In the right- hand side of (A-13). The coordinate A?' Is time-dependent, but since that time dependence shows up only In the boundaries of Integration, effectively replaced by + « In all dimensions, we can disregard It. ■ '

The cardinal assumption to be made at this point is that the velocity ;

structure is statistically Independent of the spatial structure of öe. It allows us to write t + T '

IK«^' 1^« rdt'^r^f)

*.(K;T) - Td^r' <6e(r1,)6e*(? '»e ^e

t \ ^ J 1 ^ \ i / ' ,(A-15)

t + T [t + T -i

iK- fdt1 uct') y .

The space dependence of U has not been written down In the last form because an ensemble average over space and time Is Implied by the brackets. It is also plausible that U(t') does not vary locally very much vlth t' In a time interval T £ 1 sec, or T less than soige other short time In which *4(K;T) IS essentially zero. Furthermore, velocity U Is most likely to consist of a steady wind com- ponent and a random component. I.e., Its distribution is probably adequately described by a Gaussian with non-zero mean. These two statements imply, i

<exp iK- fdt' ikt1)

(A-16)

iK,ünT 2 ■) 2 e 0 exp(-2K^AU^ T ),

± Here, U0 is the mean compnent, and we note that only its component Uo. in a plane normal to the propagation direction occurs in the first exponential. Likewise, AU-j is the variance of wind velocity in this plane. If the Wind variation is Isotropie, then it follows that AUT

2 - (2/3)AU2. Finally, all this is summarized by inserting it into (A-16) which is Inserted in1 turn into (A-15) to obtain

! I I

± ± I iK,UTT A 2 9 2

$4(K;T) - e exp(- -j KWT ) $00 , (A-17)

This set of assumptions leading to (A-17) is also known as Favre's hypothesis[8].

y

i

28 i !

!

I I

I

I

REFERENCES

1. D. A. de Wolf,: Effects of Turbulence Instabiiities on Laser Propagation, RCA Laboratories, First Quarterly Report to Rome Air Development Center,

i Griffiss AFB, New Vork, RADC-TR-71-249 (October 1971).

2. A. D. Varvatsis and M. I. Sancer, Can., J. Phys. 49, 1233 (1971).

3. R. F. Lutomirski and H. T. Yura, Appl. Opt. 10, 1652 (1971).

4. R. N. Bracewell, Handbook der Physik 54, 50 (1962).

5. J. C,. Wyngaard, Yi. Izumi, and S. A. Collins, Jr., J. Opt. Soc. Am. 61, 1646 (1971). i , :

6. P. J. Titterton, J. Opt. Soc. Am. 60, 417 (1970). i i ' '

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